Where am I wrong (submitted answer is not correct)
Thanks, let me know if anything needs more explanatioin.

Firstly, the associated eigenvector is not
1
2i
It is
1
-1-i.

Now, letting your eigenvalue be , and your eigenvector to be , you have that

Here Your is zero, so he exponentials become one. Your
Hence, substituting your values in you get
Using x(0) =-1, y(0) =-1

Can you work through all the steps? I would write it in matrix form, but am unable to using latex, I don't know how. And am too busy to hunt down how. So I called the top line of the Matrix X(t) x(t) and the bottom line y(t). I have also skipped a fair amount of the calculations, but hopefully you will be able to work your way through yourself.